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If the code and the comments disagree, then both are probably wrong. -- Norm Schryer


tech / sci.math / Re: Natural numbers and vases

SubjectAuthor
* Natural numbers and vasesWilliam
+- Re: Natural numbers and vasesRoss A. Finlayson
+* Re: Natural numbers and vasesWM
|+* Re: Natural numbers and vasesWilliam
||`* Re: Natural numbers and vasesWM
|| +- Re: Natural numbers and vasessergio
|| `* Re: Natural numbers and vasesWilliam
||  `* Re: Natural numbers and vasesWM
||   +- Re: Natural numbers and vasessergio
||   `* Re: Natural numbers and vasesWilliam
||    `* Re: Natural numbers and vasesWM
||     +- Re: Natural numbers and vasessergio
||     `* Re: Natural numbers and vasesWilliam
||      +- Re: Natural numbers and vasesFritz Feldhase
||      `* Re: Natural numbers and vasesWM
||       +* Re: Natural numbers and vasesWilliam
||       |+* Re: Natural numbers and vasesGus Gassmann
||       ||`* Re: Natural numbers and vasesFritz Feldhase
||       || `- Re: Natural numbers and vasesGus Gassmann
||       |`* Re: Natural numbers and vasesWM
||       | +- Re: Natural numbers and vasesGus Gassmann
||       | +* Re: Natural numbers and vasesWilliam
||       | |+* Re: Natural numbers and vasesWM
||       | ||+* Re: Natural numbers and vasesWilliam
||       | |||`* Re: Natural numbers and vasesWM
||       | ||| `* Re: Natural numbers and vasesWilliam
||       | |||  +* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |`- Re: Natural numbers and vasessergio
||       | |||  +* Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+- Re: Natural numbers and vasesFromTheRafters
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  ||`- Re: Natural numbers and vasesChris M. Thomasson
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+* Re: Natural numbers and vasesJim Burns
||       | |||  |||`* Re: Natural numbers and vasesWM
||       | |||  ||| +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| |`* Re: Natural numbers and vasesWM
||       | |||  ||| | +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| | |`* Re: Natural numbers and vasesWM
||       | |||  ||| | | +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| | | |`* Re: Natural numbers and vasesWM
||       | |||  ||| | | | `- Re: Natural numbers and vasessergio
||       | |||  ||| | | `- Re: Natural numbers and vasessergio
||       | |||  ||| | `- Re: Natural numbers and vasessergio
||       | |||  ||| `- Re: Natural numbers and vasessergio
||       | |||  ||`* Re: Natural numbers and vasessergio
||       | |||  || `- Re: Natural numbers and vasesChet Hirasi
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  ||`- Re: Natural numbers and vasesChet Hirasi
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`* Re: Natural numbers and vasessergio
||       | |||  || `* Re: Natural numbers and vasesWM
||       | |||  ||  `* Re: Natural numbers and vasessergio
||       | |||  ||   +* Re: Natural numbers and vasessergio
||       | |||  ||   |`* Re: Natural numbers and vasesFredJeffries
||       | |||  ||   | `- Re: Natural numbers and vasessergio
||       | |||  ||   `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||    `* Re: Natural numbers and vasessergio
||       | |||  ||     +* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||     |`- Re: Natural numbers and vasessergio
||       | |||  ||     `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||      `* Re: Natural numbers and vasessergio
||       | |||  ||       `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||        +* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||        |`- Re: Natural numbers and vasessergio
||       | |||  ||        `- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+- Re: Natural numbers and vasessergio
||       | |||  ||`* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  || +* Re: Natural numbers and vasesFromTheRafters
||       | |||  || `* Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |`* Re: Natural numbers and vasesWM
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  `- Re: Natural numbers and vaseszelos...@gmail.com
||       | ||`- Re: Natural numbers and vasessergio
||       | |`* Re: Natural numbers and vasesWM
||       | +- Re: Natural numbers and vasessergio
||       | `* Re: Natural numbers and vasesJim Burns
||       `- Re: Natural numbers and vasessergio
|+- Re: Natural numbers and vasesRoss A. Finlayson
|`- Re: Natural numbers and vasesFritz Feldhase
+* Re: Natural numbers and vasesChris M. Thomasson
+* Re: Natural numbers and vasesMostowski Collapse
`- Re: Natural numbers and vasesSocratis T.n.p.

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Re: Natural numbers and vases

<43aa1de3-a20e-26ad-a3e6-21d3016d5163@att.net>

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https://www.novabbs.com/tech/article-flat.php?id=100329&group=sci.math#100329

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Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail
From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 17 May 2022 10:58:52 -0400
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 by: Jim Burns - Tue, 17 May 2022 14:58 UTC

On 5/17/2022 9:21 AM, WM wrote:
> Jim Burns schrieb
> am Montag, 16. Mai 2022 um 22:06:48 UTC+2:

>>>> We know that,
>>>> for each index k,
>>>
>>> Please talk to the topic.

>> We do not step through fractions.
>> We do not step through indices.
>
> Wrong.
> Abzählen is basic to set theory.
> It means going through the natural numbers.

This is how we know what we know
_for each index k_ and so on.

Among what we know there are claims which describe
a fraction, statements true of a fraction
no matter which fraction is referred to.

These claims are _complete_ in the sense that
they are true of _each thing we are talking about_
when it's fractions we are talking about.
Completely true.

Starting from these initial claims true of
each fraction, which is to say: _completely true_
we can prove _further_ claims true of
each fraction, which is to say: _completely true_

These completely true claims do not _become_ true.
They _are_ true.
Stepping through fractions and
stepping through indices are irrelevant to
how we know they are completely true.

>> For example,
>> start Bob at 1
>> OXXXX...
>>
> ...
>> Thus,
>> after all swaps,
>> Bob is not at a place, any place.
>
> He is at a dark place with blurred border.

Bob is not at a place without a swap into it.
Otherwise, how did he get there?

Moreover,
for any BEFORE and AFTER of Bob's places,
Bob gets from BEFORE to AFTER,
and he does it in one swap.
Bob only moves in swaps.

Bob is not at any place without,
for each BEFORE and AFTER =< that place,
some j ends BEFORE and j+1 begins AFTER.
Otherwise, how did he get from one to the other?

That's _completely true_ of Bob's places.
Its completely-true-ness is how we know that
Bob is not at a dark place.

> Otherwise, in a set with only well-defined lit
> elements, there would have been a last place
> where he was.

<WM>
>
> For each place with a swap into it,
> there is a swap out of it.
>
</WM>

>> The places in {1,2,3,...} match
>> the places in {2,3,4,...}.
>
> That is wrong because
> the upper set has one more element.

⟨1,2⟩, ⟨2,3⟩, ⟨3,4⟩, ⟨4,5⟩, ...

Re: Natural numbers and vases

<t60pcq$f64$1@dont-email.me>

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 17 May 2022 11:26:34 -0700
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 by: Chris M. Thomasson - Tue, 17 May 2022 18:26 UTC

On 5/17/2022 7:19 AM, Fritz Feldhase wrote:
> On Tuesday, May 17, 2022 at 3:22:01 PM UTC+2, WM wrote:
>> Jim Burns schrieb am Montag, 16. Mai 2022 um 22:06:48 UTC+2:
>>> On 5/16/2022 12:59 PM, WM wrote:
>>>
>>> We do not step through fractions.
>>> We do not step through indices.
>>>
>> Wrong. Abzählen [counting] is basic to set theory. It means going through the natural numbers.
>
> Indeed! Note that Chuck Norris counted to infinity - twice.

In a roundhouse to the face. LOL! ;^)

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Tue, 17 May 2022 20:42 UTC

FromTheRafters schrieb am Dienstag, 17. Mai 2022 um 16:15:45 UTC+2:
> WM formulated the question :
> > FromTheRafters schrieb am Dienstag, 17. Mai 2022 um 09:09:37 UTC+2:
> >> WM used his keyboard to write :
> >
> >>>>> At which k? k is not a number and therefore not finite.
> >>>> k is an element of N.
> >>>
> >>> Which one is it?
> >> No matter, every element of N is finite.
> >
> > You said k is an elements of |N. Which is it?
> It doesn't matter!

If it is an element we should be able to know which element it is.

> It is a feature of each and every element.

There is an element which is a feature of ever element? Not in |N!

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Tue, 17 May 2022 20:46 UTC

William schrieb am Dienstag, 17. Mai 2022 um 15:57:47 UTC+2:
> > the integer fractions have to supply these indexes.
> The set of integer fractions, like the set N_p, like any Peano set, has cadinality aleph_0. There is no problem with this set supplying a set of indexes with cardinality aleph_0 You do not like the idea of a bijection between a set of cardinality aleph_0 and a proper subset, but this is not a contradiction it is something you do not like.

This is a contradiction: Every index given to a fraction is taken form an originally indexed integer fraction. Therefore the set of not indexed fractions does never decrease.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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 by: sergi o - Tue, 17 May 2022 21:41 UTC

On 5/17/2022 3:46 PM, WM wrote:
> William schrieb am Dienstag, 17. Mai 2022 um 15:57:47 UTC+2:
>>> the integer fractions have to supply these indexes.
>> The set of integer fractions, like the set N_p, like any Peano set, has cadinality aleph_0. There is no problem with this set supplying a set of indexes with cardinality aleph_0 You do not like the idea of a bijection between a set of cardinality aleph_0 and a proper subset, but this is not a contradiction it is something you do not like.
>
> This is a contradiction: Every index given to a fraction is taken form an originally indexed integer fraction.

yes, that is called a "hot swapparoo",

I gave the index 5 to 1/2, but I had to get the 5 from the originally 3/17. (IAW your statement above)

SO, 5 got swapparooed from 3/17 to 1/2, but now 3/17 is not indexed!!

I could have "swapparoo with cloning", where I clone the 5 from the 3/17, and paste the new clone 5 on the 1/2, and the other (clone) 5 pasted onto the
3/17.

it is called "hot", cause you are looking at the number, now suddenly instantated, and you goof on it, by pasting a "used index" on it.

> Therefore the set of not indexed fractions does never decrease.

you should use stickers instead of paste, its is a lot faster.

and get New Indexes, never reuse an index, a 'used index' is just nasty!

>
> Regards, WM
>

Re: Natural numbers and vases

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 17 May 2022 16:44:38 -0500
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 by: sergi o - Tue, 17 May 2022 21:44 UTC

On 5/17/2022 3:42 PM, WM wrote:
> FromTheRafters schrieb am Dienstag, 17. Mai 2022 um 16:15:45 UTC+2:
>> WM formulated the question :
>>> FromTheRafters schrieb am Dienstag, 17. Mai 2022 um 09:09:37 UTC+2:
>>>> WM used his keyboard to write :
>>>
>>>>>>> At which k? k is not a number and therefore not finite.
>>>>>> k is an element of N.
>>>>>
>>>>> Which one is it?
>>>> No matter, every element of N is finite.
>>>
>>> You said k is an elements of |N. Which is it?
>> It doesn't matter!
>
> If it is an element we should be able to know which element it is.

why do you require indexing of all sets of elements ?

>
>> It is a feature of each and every element.
>
> There is an element which is a feature of ever element? Not in |N!

wrong, its feature is that it is an element of |N

>
> Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Tue, 17 May 2022 22:39 UTC

On Tuesday, May 17, 2022 at 5:46:07 PM UTC-3, WM wrote:

> This is a contradiction:

Piffle. It is trivial to see that there is a bijection from any Peano set, like the integer fractions, to a proper subset. The existence of a Peoano set is guaranteed by the Axiom of Infinity. You continue to argue that the axiom of infinity leads to results you think "repugnant to the nature" of mathematicas.

--
William Hughes

Re: Natural numbers and vases

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 17 May 2022 19:07:56 -0400
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 by: Jim Burns - Tue, 17 May 2022 23:07 UTC

On 5/17/2022 4:42 PM, WM wrote:
> FromTheRafters schrieb
> am Dienstag, 17. Mai 2022 um 16:15:45 UTC+2:
>> WM formulated the question :
>>> FromTheRafters schrieb
> am Dienstag, 17. Mai 2022 um 09:09:37 UTC+2:
>>>> WM used his keyboard to write :

>>>>>>> At which k?
>>>>>>> k is not a number and therefore not finite.
>>>>>>
>>>>>> k is an element of N.
>>>>>
>>>>> Which one is it?
>>>>
>>>> No matter, every element of N is finite.
>>>
>>> You said k is an elements of |N. Which is it?
>>
>> It doesn't matter!
>
> If it is an element
> we should be able to know which element it is.

If it is an element
we should be able to know things about it which
are true of each and every element.

>> It is a feature of each and every element.
>
> There is an element which is a feature of
> ever element? Not in |N!

When we state what we mean by "natural number",
that is true of each natural number.
A _completely true_ claim.

There are many ways to do state that.
I'm pretty sure that all of them rule out
dark numbers as things we are talking about.
It seems to me that what you (WM) want is
a description of natural numbers vague enough
to permit your dark numbers to be included.

And, if it isn't vague enough for that,
your plan is to pout and call us matheologians.

Re: Natural numbers and vases

<t61en2$2a6$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 17 May 2022 17:30:23 -0700
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 by: FromTheRafters - Wed, 18 May 2022 00:30 UTC

WM used his keyboard to write :
> FromTheRafters schrieb am Dienstag, 17. Mai 2022 um 16:15:45 UTC+2:
>> WM formulated the question :
>>> FromTheRafters schrieb am Dienstag, 17. Mai 2022 um 09:09:37 UTC+2:
>>>> WM used his keyboard to write :
>>>>>>> At which k? k is not a number and therefore not finite.
>>>>>> k is an element of N.
>>>>>
>>>>> Which one is it?
>>>> No matter, every element of N is finite.
>>>
>>> You said k is an elements of |N. Which is it?
>> It doesn't matter!
>
> If it is an element we should be able to know which element it is.

Not that crap again. :(

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 12:52 UTC

horand....@gmail.com schrieb am Dienstag, 17. Mai 2022 um 23:27:41 UTC+2:
> On Tuesday, 17 May 2022 at 17:46:07 UTC-3, WM wrote:
>
> > This is a contradiction: Every index given to a fraction is taken form an originally indexed integer fraction. Therefore the set of not indexed fractions does never decrease.
> Will wonders never cease? Taking a finite number of integers from an infinite set leaves an infinite set!

Of course! But there is a slight difference. Taking a number of integers from an infinite set increases the taken set and raises the impression of some fools that finally the infinite set could be exhausted nevertheless. Contrary to that, my argument does not increase the taken set in the least such that not-exhausting of the infinite set is matter of fact.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 12:57 UTC

Jim Burns schrieb am Dienstag, 17. Mai 2022 um 16:59:03 UTC+2:

> These completely true claims do not _become_ true.
> They _are_ true.

This one, for instance:

We check the number of indexes by bijecting them with the fractions of the first column (we could use every other column or line as well).
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
....
When applying the indexes for indexing fractions according to m/n gets the index k from
k = (m + n - 1)(m + n - 2)/2 + m
with the result
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
then the integer fractions have to supply these indexes. They are stripped off these indexes.

No? This one is an exception?

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 13:01 UTC

William schrieb am Mittwoch, 18. Mai 2022 um 00:39:14 UTC+2:
> On Tuesday, May 17, 2022 at 5:46:07 PM UTC-3, WM wrote:
>
> > This is a contradiction:
>
> Piffle. It is trivial to see that there is a bijection from any Peano set, like the integer fractions, to a proper subset.

All numbers you get by induction have ℵo successors before ω:
∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
They cannot be exhausted, because they remain always there.

But the set of all natural numbers exhausts also these successors:
|ℕ \ {1, 2, 3, ...}| = 0
or
{0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.

> The existence of a Peoano set is guaranteed by the Axiom of Infinity. You continue to argue that the axiom of infinity leads to results you think "repugnant to the nature" of mathematicas.

Is this true?
∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
Can these elements be gathered into a set? Do they there change their proerties and strip off the successors between every element and omega?

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 13:04 UTC

Jim Burns schrieb am Mittwoch, 18. Mai 2022 um 01:08:06 UTC+2:
> On 5/17/2022 4:42 PM, WM wrote:

> > If it is an element
> > we should be able to know which element it is.
>
> If it is an element
> we should be able to know things about it which
> are true of each and every element.

That does not answer the question which element it is. k has been claimed to be an element.

> When we state what we mean by "natural number",
> that is true of each natural number.
> A _completely true_ claim.

But whe we state that k is an element, then we have to answer whether or not it is less than 10.
>
> There are many ways to do state that.
> I'm pretty sure that all of them rule out
> dark numbers as things we are talking about.
> It seems to me that what you (WM) want is
> a description of natural numbers vague enough
> to permit your dark numbers to be included.
>
All numbers you get by induction have ℵo successors before ω:
∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
They cannot be exhausted, because they remain always there.

But the set of all natural numbers exhausts also these successors:
|ℕ \ {1, 2, 3, ...}| = 0
or
{0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 13:08 UTC

horand....@gmail.com schrieb am Mittwoch, 18. Mai 2022 um 15:04:54 UTC+2:
> On Wednesday, 18 May 2022 at 09:52:21 UTC-3, WM wrote:

> > Of course! But there is a slight difference. Taking a number of integers from an infinite set increases the taken set and raises the impression of some fools that finally the infinite set could be exhausted nevertheless. Contrary to that, my argument does not increase the taken set in the least such that [...]
>
> ...the only thing that is glaringly obvious is that the professor hasn't got a clue about limits.

First: Limits are not relevant when enumerating elements. Second: The limit of a constant set is a constant set.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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 by: sergi o - Wed, 18 May 2022 13:59 UTC

On 5/18/2022 7:52 AM, WM wrote:
> horand....@gmail.com schrieb am Dienstag, 17. Mai 2022 um 23:27:41 UTC+2:
>> On Tuesday, 17 May 2022 at 17:46:07 UTC-3, WM wrote:
>>
>>> This is a contradiction: Every index given to a fraction is taken form an originally indexed integer fraction. Therefore the set of not indexed fractions does never decrease.
>> Will wonders never cease? Taking a finite number of integers from an infinite set leaves an infinite set!
>
> Of course! But there is a slight difference. Taking a number of integers from an infinite set increases the taken set

No. The taken set is a fixed finite set. Your increase is deception.

> and raises the impression of some fools that finally the infinite set could be exhausted nevertheless.

you are the one that "exhausts" infinite sets.

> Contrary to that, my argument does not increase the taken set in the least such that not-exhausting of the infinite set is matter of fact.

your argument is moot, a diversion.

>
> Regards, WM

Re: Natural numbers and vases

<t62ubq$1dvp$1@gioia.aioe.org>

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 by: sergi o - Wed, 18 May 2022 14:03 UTC

On 5/18/2022 8:08 AM, WM wrote:
> horand....@gmail.com schrieb am Mittwoch, 18. Mai 2022 um 15:04:54 UTC+2:
>> On Wednesday, 18 May 2022 at 09:52:21 UTC-3, WM wrote:
>
>>> Of course! But there is a slight difference. Taking a number of integers from an infinite set increases the taken set and raises the impression of some fools that finally the infinite set could be exhausted nevertheless. Contrary to that, my argument does not increase the taken set in the least such that [...]
>>
>> ...the only thing that is glaringly obvious is that the professor hasn't got a clue about limits.
>
> First: Limits are not relevant when enumerating elements.

then why do you limit your swaparoofest to finite ?

> Second: The limit of a constant set is a constant set.

Define a constant set.
do you mean a set that does not change with time ?
or do you mean a set of constants
or do you mean sets with the same number of elements ?

>
> Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Wed, 18 May 2022 14:27 UTC

No *element* of N_p is infinite. The *set* N_p is infinite.

--
William Hughes

Re: Natural numbers and vases

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From: inva...@invalid.com (sergi o)
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Subject: Re: Natural numbers and vases
Date: Wed, 18 May 2022 09:40:26 -0500
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 by: sergi o - Wed, 18 May 2022 14:40 UTC

On 5/18/2022 7:57 AM, WM wrote:
> Jim Burns schrieb am Dienstag, 17. Mai 2022 um 16:59:03 UTC+2:
>
>> These completely true claims do not _become_ true.
>> They _are_ true.
>
>
>
> We enumerate the rationals by bijecting them with the natural numbers
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
> When applying the indexes for indexing fractions according to m/n gets the index k from
> k = (m + n - 1)(m + n - 2)/2 + m
> with the result
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>
>
> Regards, WM
>

above corrected.

Re: Natural numbers and vases

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From: inva...@invalid.com (sergi o)
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Subject: Re: Natural numbers and vases
Date: Wed, 18 May 2022 09:46:13 -0500
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 by: sergi o - Wed, 18 May 2022 14:46 UTC

On 5/18/2022 8:04 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 18. Mai 2022 um 01:08:06 UTC+2:
>> On 5/17/2022 4:42 PM, WM wrote:
>
>>> If it is an element
>>> we should be able to know which element it is.
>>
>> If it is an element
>> we should be able to know things about it which
>> are true of each and every element.
>
> That does not answer the question which element it is. k has been claimed to be an element.

you are just bullshitting again. Here is your equation;

∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

That does not answer the question which element n is. n has been claimed to be an element.

which one is n ? or is your equation deeply flawed ?

>
>> When we state what we mean by "natural number",
>> that is true of each natural number.
>> A _completely true_ claim.
>
> But whe we state that k is an element, then we have to answer whether or not it is less than 10.

you put on that restriction k < 10.
I stated that k ∈ ℕ without restriction, only that it is an element of the set of natural numbers.

>>
>> There are many ways to do state that.
>> I'm pretty sure that all of them rule out
>> dark numbers as things we are talking about.
>> It seems to me that what you (WM) want is
>> a description of natural numbers vague enough
>> to permit your dark numbers to be included.
>>
> All numbers you get by induction have ℵo successors before ω:
> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> They cannot be exhausted, because they remain always there.

so, what is n ?

"then we have to answer whether or not it is less than 10".

deception.

> Regards, WM

Re: Natural numbers and vases

<t635eo$dgc$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases
Date: Wed, 18 May 2022 09:04:37 -0700
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 by: FromTheRafters - Wed, 18 May 2022 16:04 UTC

It happens that WM formulated :
> horand....@gmail.com schrieb am Dienstag, 17. Mai 2022 um 23:27:41 UTC+2:
>> On Tuesday, 17 May 2022 at 17:46:07 UTC-3, WM wrote:
>>
>>> This is a contradiction: Every index given to a fraction is taken form an
>>> originally indexed integer fraction. Therefore the set of not indexed
>>> fractions does never decrease.
>> Will wonders never cease? Taking a finite number of integers from an
>> infinite set leaves an infinite set!
>
> Of course! But there is a slight difference. Taking a number of integers from
> an infinite set increases the taken set and raises the impression of some
> fools that finally the infinite set could be exhausted nevertheless. Contrary
> to that, my argument does not increase the taken set in the least such that
> not-exhausting of the infinite set is matter of fact.

What?

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: horand.g...@gmail.com (Gus Gassmann)
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 by: Gus Gassmann - Wed, 18 May 2022 16:24 UTC

On Wednesday, 18 May 2022 at 13:04:51 UTC-3, FromTheRafters wrote:
> It happens that WM formulated :
> > horand....@gmail.com schrieb am Dienstag, 17. Mai 2022 um 23:27:41 UTC+2:
> >> On Tuesday, 17 May 2022 at 17:46:07 UTC-3, WM wrote:
> >>
> >>> This is a contradiction: Every index given to a fraction is taken form an
> >>> originally indexed integer fraction. Therefore the set of not indexed
> >>> fractions does never decrease.
> >> Will wonders never cease? Taking a finite number of integers from an
> >> infinite set leaves an infinite set!
> >
> > Of course! But there is a slight difference. Taking a number of integers from
> > an infinite set increases the taken set and raises the impression of some
> > fools that finally the infinite set could be exhausted nevertheless. Contrary
> > to that, my argument does not increase the taken set in the least such that
> > not-exhausting of the infinite set is matter of fact.
> What?

Well, in a sense he is right: Since he is too dense to even understand what the task is, he never starts and pretends that that has done the job. It is clear that trying to talk sense into him is a waste of breath (or electrons, as the case may be), but it does give me the opportunity to vent my frustrations and hurl gratuitous insults at him --- which he richly deserves!

Re: Natural numbers and vases

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 18 May 2022 14:15:15 -0400
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 by: Jim Burns - Wed, 18 May 2022 18:15 UTC

On 5/18/2022 8:57 AM, WM wrote:
> Jim Burns schrieb
> am Dienstag, 17. Mai 2022 um 16:59:03 UTC+2:

>> These completely true claims do not _become_ true.
>> They _are_ true.
>
> This one, for instance:
>
> We check the number of indexes by bijecting them
> with the fractions of the first column (we could use
> every other column or line as well).
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> When applying the indexes for indexing fractions
> according to m/n gets the index k from
>
> k = (m + n - 1)(m + n - 2)/2 + m
>
> with the result
>
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>
> then the integer fractions have to supply these indexes.
> They are stripped off these indexes.
>
> No? This one is an exception?

Some claims are true of each natural number.
Some claims are not true of each natural number.

We can reason from the first kind of claim to
further claims also true of each natural number.

I don't know what you mean by "exception".
Do you call a triangle an exception to being a square?

----
Consider the first column of The Matrix.
k/1

1/1
2/1
3/1
4/1
5/1
6/1
7/1
8/1
9/1
10/1
11/1
....

For each numerator k,
there is a _triangle number_ k*(k+1)/2
k/1 k*(k+1)/2

1/1 1
2/1 3
3/1 6
4/1 10
5/1 15
6/1 21
7/1 28
8/1 36
9/1 45
10/1 55
11/1 66
....

For each numerator k,
there is a last triangle number < k
there is m = k - (last triangle num < k)
k/1 m

1/1 0 1
2/1 1 1
3/1 1 2
4/1 3 1
5/1 3 2
6/1 3 3
7/1 6 1
8/1 6 2
9/1 6 3
10/1 6 4
11/1 10 1
....

For each numerator k,
there is a last triangle number < k
there is s = the index of (last triangle num < k)
there is n = s + 2 - m
k/1 s n

1/1 0 0 1
2/1 1 1 2
3/1 1 1 1
4/1 3 2 3
5/1 3 2 2
6/1 3 2 1
7/1 6 3 4
8/1 6 3 3
9/1 6 3 2
10/1 6 3 1
11/1 10 4 5
....

For each numerator k,
there is a fraction m/n
k/1 m/n

1/1 1/1
2/1 1/2
3/1 2/1
4/1 1/3
5/1 2/2
6/1 3/1
7/1 1/4
8/1 2/3
9/1 3/2
10/1 4/1
11/1 1/5
....

No exceptions.

----
Consider The Matrix.
1/1 1/2 1/3 1/4 1/5 ...
2/1 2/2 2/3 2/4 2/5 ...
3/1 3/2 3/3 3/4 3/5 ...
4/1 4/2 4/3 4/4 4/5 ...
5/1 5/2 5/3 5/4 5/5 ...
....

For each fraction m/n,
there is s = m + n
2 3 4 5 6 ...
3 4 5 6 7 ...
4 5 6 7 8 ...
5 6 7 8 9 ...
6 7 8 9 10 ...
....

For each fraction m/n,
there is (s-1)*(s-2)/2
0 1 3 6 10 ...
1 3 6 10 15 ...
3 6 10 15 21 ...
6 10 15 21 28 ...
10 15 21 28 36 ...
....

For each fraction m/n,
there is k = m + (s-2)*(s-2)/2
1 2 4 7 11 ...
3 5 7 12 17 ...
6 8 13 18 24 ...
10 14 19 25 32 ...
15 20 26 33 41 ...
....

No exceptions.

----
> When applying the indexes for indexing fractions
> according to m/n gets the index k from
>
> k = (m + n - 1)(m + n - 2)/2 + m
>
> with the result
>
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>
> then the integer fractions have to supply these indexes.
> They are stripped off these indexes.
>
> No? This one is an exception?

What are you asking if it might be an exception?
An exception to what?

Re: Natural numbers and vases

<t63eri$1e6t$1@gioia.aioe.org>

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 18 May 2022 13:45:04 -0500
Organization: Aioe.org NNTP Server
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 by: sergi o - Wed, 18 May 2022 18:45 UTC

On 5/18/2022 1:15 PM, Jim Burns wrote:
> On 5/18/2022 8:57 AM, WM wrote:
>> Jim Burns schrieb
>> am Dienstag, 17. Mai 2022 um 16:59:03 UTC+2:
>
>>> These completely true claims do not _become_ true.
>>> They _are_ true.
>>
>> This one, for instance:
>>
>> We check the number of indexes by bijecting them
>> with the fractions of the first column (we could use
>> every other column or line as well).
>>
>> 1/1, 1/2, 1/3, 1/4, ...
>> 2/1, 2/2, 2/3, 2/4, ...
>> 3/1, 3/2, 3/3, 3/4, ...
>> 4/1, 4/2, 4/3, 4/4, ...
>> 5/1, 5/2, 5/3, 5/4, ...
>> ...
>>
>> When applying the indexes for indexing fractions
>> according to m/n gets the index k from
>>
>> k = (m + n - 1)(m + n - 2)/2 + m
>>
>> with the result
>>
>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>>
>> then the integer fractions have to supply these indexes.
>> They are stripped off these indexes.
>>
>> No? This one is an exception?
>
> Some claims are true of each natural number.
> Some claims are not true of each natural number.
>
> We can reason from the first kind of claim to
> further claims also true of each natural number.
>
> I don't know what you mean by "exception".
> Do you call a triangle an exception to being a square?
>
> ----
> Consider the first column of The Matrix.
> k/1
>
> 1/1
> 2/1
> 3/1
> 4/1
> 5/1
> 6/1
> 7/1
> 8/1
> 9/1
> 10/1
> 11/1
> ...
>
> For each numerator k,
> there is a _triangle number_ k*(k+1)/2
> k/1  k*(k+1)/2
>
> 1/1  1
> 2/1  3
> 3/1  6
> 4/1  10
> 5/1  15
> 6/1  21
> 7/1  28
> 8/1  36
> 9/1  45
> 10/1 55
> 11/1 66
> ...
>
> For each numerator k,
> there is a last triangle number < k
> there is m  =  k - (last triangle num < k)
> k/1     m
>
> 1/1  0  1
> 2/1  1  1
> 3/1  1  2
> 4/1  3  1
> 5/1  3  2
> 6/1  3  3
> 7/1  6  1
> 8/1  6  2
> 9/1  6  3
> 10/1 6  4
> 11/1 10 1
> ...
>
> For each numerator k,
> there is a last triangle number < k
> there is s = the index of (last triangle num < k)
> there is n  =  s + 2 - m
> k/1     s  n
>
> 1/1  0  0  1
> 2/1  1  1  2
> 3/1  1  1  1
> 4/1  3  2  3
> 5/1  3  2  2
> 6/1  3  2  1
> 7/1  6  3  4
> 8/1  6  3  3
> 9/1  6  3  2
> 10/1 6  3  1
> 11/1 10 4  5
> ...
>
> For each numerator k,
> there is a fraction m/n
> k/1  m/n
>
> 1/1  1/1
> 2/1  1/2
> 3/1  2/1
> 4/1  1/3
> 5/1  2/2
> 6/1  3/1
> 7/1  1/4
> 8/1  2/3
> 9/1  3/2
> 10/1  4/1
> 11/1  1/5
> ...
>
> No exceptions.
>
> ----
> Consider The Matrix.
> 1/1 1/2 1/3 1/4 1/5 ...
> 2/1 2/2 2/3 2/4 2/5 ...
> 3/1 3/2 3/3 3/4 3/5 ...
> 4/1 4/2 4/3 4/4 4/5 ...
> 5/1 5/2 5/3 5/4 5/5 ...
> ...
>
> For each fraction m/n,
> there is s = m + n
> 2   3   4   5   6   ...
> 3   4   5   6   7   ...
> 4   5   6   7   8   ...
> 5   6   7   8   9   ...
> 6   7   8   9   10  ...
> ...
>
> For each fraction m/n,
> there is (s-1)*(s-2)/2
> 0   1   3   6   10  ...
> 1   3   6   10  15  ...
> 3   6   10  15  21  ...
> 6   10  15  21  28  ...
> 10  15  21  28  36  ...
> ...
>
> For each fraction m/n,
> there is k  =  m + (s-2)*(s-2)/2
> 1   2   4   7   11  ...
> 3   5   7   12  17  ...
> 6   8   13  18  24  ...
> 10  14  19  25  32  ...
> 15  20  26  33  41  ...
> ...
>
> No exceptions.
>
> ----
>> When applying the indexes for indexing fractions
>> according to m/n gets the index k from
>>
>> k = (m + n - 1)(m + n - 2)/2 + m
>>
>> with the result
>>
>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>>
>> then the integer fractions have to supply these indexes.
>> They are stripped off these indexes.
>>
>> No? This one is an exception?
>
> What are you asking if it might be an exception?
> An exception to what?
>
>
>

that is a very clear presentation.

Re: Natural numbers and vases

<773906b8-f18b-623a-684b-80a5179b60bb@att.net>

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https://www.novabbs.com/tech/article-flat.php?id=100430&group=sci.math#100430

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 18 May 2022 15:16:05 -0400
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 by: Jim Burns - Wed, 18 May 2022 19:16 UTC

On 5/18/2022 2:45 PM, sergi o wrote:
> On 5/18/2022 1:15 PM, Jim Burns wrote:
>> On 5/18/2022 8:57 AM, WM wrote:

>>> When applying the indexes for indexing fractions
>>> according to m/n gets the index k from
>>>
>>> k = (m + n - 1)(m + n - 2)/2 + m
>>>
>>> with the result
>>>
>>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>>>
>>> then the integer fractions have to supply these indexes.
>>> They are stripped off these indexes.
>>>
>>> No? This one is an exception?
>>
>> Some claims are true of each natural number.
>> Some claims are not true of each natural number.
>>
>> We can reason from the first kind of claim to
>> further claims also true of each natural number.
>>
>> I don't know what you mean by "exception".
>> Do you call a triangle an exception to being a square?
>>
>> ----
>> Consider the first column of The Matrix.

>> For each numerator k,
>> there is a fraction m/n

>> No exceptions.
>>
>> ----
>> Consider The Matrix.

>> For each fraction m/n,
>> there is k  =  m + (s-2)*(s-2)/2

>> No exceptions.
>>
>> ----

>> What are you asking if it might be an exception?
>> An exception to what?

> that is a very clear presentation.

Thank you for your kind words.

I would very much like WM to read it,
but I make no prediction in that regard.

He's not my only possible reader, though.
I imagine myself putting a light in the window
for those trying to find their way out of WM's fog.

For that purpose, it's not possible to be too clear.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Thu, 19 May 2022 13:41 UTC

horand....@gmail.com schrieb am Mittwoch, 18. Mai 2022 um 18:24:37 UTC+2:
> On Wednesday, 18 May 2022 at 13:04:51 UTC-3, FromTheRafters wrote:
> > It happens that WM formulated :
> > > my argument does not increase the taken set in the least such that
> > > not-exhausting of the infinite set is matter of fact.
> > What?
> Well, in a sense he is right:

Of course. The shares of enumerated and not enumerated fractions do never change. That is the only important feature. Whether we can finish or not, is irrelevant.

Regards, WM

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